where \(\beta _{s} = \mathcal {R} T \Gamma _{\infty }/\sigma _{s}\)  is the surfactant elasticity number; \(\mathcal {R}\)  and \(T\)  are the thermodynamic ideal gas constant and temperature, respectively. The non-linear equation of state described above is valid for very dilute systems in which \(\Gamma << \Gamma _{\infty }\) . As \(\Gamma \)  increases, the equation of state yields unphysical negative values of surface tension. To circumvent this, a limiting value for \(\tilde{\sigma }\) , \(\epsilon _{\sigma } = 0.05\) , has been introduced, in accordance with [1]} and [2]}. The dimensionless groups that appear in the above equations and that characterise the system are defined as follows:
\(\begin{split}Re=\frac{\rho _lU_{a} D}{\mu _l};~Ca=\frac{\mu _{l}U_{a}}{\sigma _s};~Pe_c=\frac{U_{a} D}{D_c};~Pe_s=\frac{U_{a} D}{D_s};~ Bi=\frac{k_d D}{U_{a}}; \\ ~Da=\frac{\Gamma _\infty }{D C_{\infty }};~k=\frac{k_a C_{\infty }}{k_d},\end{split}\) 
